Exercises: Derivative function

A skater moves along a track following a trajectory whose position (in metres) is given by the function $s(t) = t^2 + 3t$, where $t$ is the time in seconds. Calculate, using the definition of derivative as a limit, the instantaneous velocity of the skater at $t = 2$ seconds.

Calculate the derivative function of the following elementary functions using differentiation rules:

1. a) $f(x) = 5x^4 - 3x^2 + 7x - 1$

1. b) $g(x) = \frac{1}{x^3} + \sqrt{x}$

1. c) $h(x) = 3e^x - 2\cos(x)$

Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 4x + 1$ at the point with abscissa $x = 3$.

Given the function $f(x) = x \times \ln(x)$, calculate its derivative $f'(x)$ applying the product rule and determine the gradient of the curve at the point where $x = e$.

An access ramp for a skatepark is being designed. The side profile of the first section follows the function $f(x) = \frac{x^2+4}{x+1}$. Calculate the derivative function $f'(x)$ to find how the incline of the ramp varies at any point along that section.

Apply the chain rule to find the derivative of the following composite functions:

1. a) $f(x) = (3x^2 - 5)^4$

1. b) $g(x) = \sin(5x + 2)$

1. c) $h(x) = e^{x^2}$

Study the differentiability of the following function at the point $x = 2$, first checking whether it is continuous at that point:

Calculate the second-order derivative (second derivative) of the function $f(x) = x^3 - 6x^2 + 9x - 2$. For which value of $x$ does this second derivative equal zero? (This point is important for studying curvature, which you will see later).

Aoife is calculating the derivative of the function $f(x) = x^{2x+1}$ for $x > 0$. On her exam, she writes the following solution:

1. a) Explain in detail the conceptual error Aoife makes and why the usual differentiation rules for power functions ($x^k$) or exponential functions ($a^x$) cannot be applied to this function.
2. b) Correct the exercise by applying the logarithmic differentiation method step by step to obtain the correct expression for $f'(x)$, leaving it simplified and expressed exclusively in terms of $x$.

We want to calculate the derivative function of $f(x) = \ln\left(\frac{e^{4x} \times (x^2 + 3)^5}{\sqrt{2x + 1}}\right)$ on the domain $x > -\frac{1}{2}$. Two approaches are proposed to solve it:

• Method A: Directly apply the chain rule for the logarithm function, $f'(x) = \frac{u'(x)}{u(x)}$, calculating $u'(x)$ using the quotient rule combined with the product and power rules.

• Method B: Apply the properties of logarithms (logarithm of a product, of a quotient, and of a power) to transform $f(x)$ into a linear combination of simple terms before differentiating.

1. a) Compare both methods with justification, assessing the difficulty of the algebraic calculation and the likelihood of making errors with the chain rule.
2. b) Differentiate the function using the method you have justified as the most efficient, and simplify the final result.

Calculate the derivative function of the following composite function, simplifying the result as much as possible:

$f(x) = \ln\left(\frac{1 + \sin(x)}{1 - \sin(x)}\right)$

Determine the values of the parameters $a$ and $b$ so that the following function is differentiable throughout its domain, paying special attention to the joining point $x = 0$:

$f(x) = \begin{cases} e^{ax} & \text{if } x \leq 0 \\ x^2 + bx + 1 & \text{if } x > 0 \end{cases}$

Given the function $f(x) = \ln(2x - 3)$, find the general expression for its nth derivative $f^{(n)}(x)$ and use it to calculate the value of the fourth derivative at the point $x = 2$.

Find the equation of the tangent line to the curve $f(x) = \frac{x^2 - 1}{x + 2}$ at the point where the function crosses the y-axis ($OY$).

A surveillance drone follows a path whose height in metres is given by the function $h(t) = \frac{1}{3}t^3 - 4t^2 + 11t + 5$, where $t$ is the time in seconds ($0 \leq t \leq 10$).

• a) Calculate the velocity function $v(t)$ and the acceleration function $a(t)$.

• b) At what times is the drone instantaneously at rest (velocity equal to zero)?

Use the technique of logarithmic differentiation to find the derivative of the following non-algebraic power function:

$f(x) = (x^2 + 1)^{\cos(x)}$

Study the differentiability of the function $f(x) = |x^2 - 4x + 3|$ at the points $x = 1$ and $x = 3$.

1. a) Study whether $f$ is continuous at $x = 1$ and at $x = 3$.

1. b) Calculate, if they exist, the one-sided derivatives of $f$ at $x = 1$ and at $x = 3$.

1. c) State whether $f$ is differentiable at those points.

Given the equation of the curve implicitly defined by $x^3 + y^3 - 3xy = 0$ (known as the Folium of Descartes), calculate the value of the gradient of the tangent line to the curve at the point $( \frac{3}{2}, \frac{3}{2} )$.

According to data from EirGrid, on a winter day the national electrical power demand $P(t)$, expressed in gigawatts (GW), can be approximated between $06:00$ and $14:00$ hours ($t \in [6, 14]$, where $t$ is the hour of the day) by the cubic function:

1. a) Obtain the algebraic expression for $P'(t)$, which represents the instantaneous rate of change of electricity demand, and state its units of measurement with justification.
2. b) Determine the time intervals in which national consumption is increasing and those in which it is decreasing within that time range.
3. c) Find the exact time of the morning at which the peak demand is reached and what the power consumed is at that instant. Analytically justify, using the first or second derivative test, why said value is a relative maximum and not an inflection point.

Consider the function with variable base and exponent $f(x) = x^{\sin(x)}$, defined for all $x > 0$.

1. a) Estimate: Without performing algebraic derivatives, qualitatively analyze the behavior of the base $u(x) = x$ and of the exponent $v(x) = \sin(x)$ near $x = \frac{\pi}{2}$. Taking into account that the sine function reaches a relative maximum at $\frac{\pi}{2}$ (so its instantaneous rate of change is zero at that point), estimate with justification whether $f(x)$ will be increasing or decreasing at $x = \frac{\pi}{2}$ and deduce what sign you expect for $f'\left(\frac{\pi}{2}\right)$.
2. b) Calculate: Find the derivative function $f'(x)$ using logarithmic differentiation and calculate the exact numerical value of $f'\left(\frac{\pi}{2}\right)$.
3. c) Check: Assess whether the obtained value confirms your previous qualitative estimate and geometrically interpret the slope of the tangent line at that point.

Consider the piecewise function $f: \mathbb{R} \to \mathbb{R}$ defined by the following expression:

Determine the values of the parameters $a, b \in \mathbb{R}$ so that the function is differentiable throughout its domain. Once found, discuss whether the derivative function $f'(x)$ is, in turn, continuous at $x = 2$.

Using only the definition of the derivative as a limit (difference quotient), calculate the derivative function of $f(x) = \sqrt{2x^2 + 1}$. Then, use the result to find the equation of the tangent line to the curve at the point $P$ with abscissa $x = 2$.

The figure shows the graph of $f$ and the position of the point $P$.

Given the curve defined by the implicit equation $x^3 + y^3 - 3xy = 0$ (known as the Folium of Descartes), answer the following:

1. a) Obtain the expression for the derivative $y'$ in terms of $x$ and $y$ using implicit differentiation.
2. b) Find the points on the curve where the tangent line is horizontal.
3. c) Determine the equation of the normal line to the curve at the point $( \frac{3}{2}, \frac{3}{2} )$.

Use the technique of logarithmic differentiation to calculate the derivative function of:

State with reasoning the domain of definition of the function $f(x)$ and that of its derivative function $f'(x)$. Is there any point in the interval $(0, \pi/2)$ where the gradient of the tangent line is zero?

Given the function $f(x) = \frac{1}{x^2 - 3x + 2}$:

1. a) Decompose the function into partial fractions.
2. b) Using the above decomposition, find a general formula for the $n$th derivative, $f^{(n)}(x)$, for any $n \in \mathbb{N}$.
3. c) Prove the validity of your formula for every $n \in \mathbb{N}$ by mathematical induction, explicitly checking the first three orders of differentiation.

A surveillance drone follows a flight path along a straight axis. Its position (in decameters) relative to the control center is given by the function $s(t) = t^2 \times e^{-t/4}$, where $t$ is the time in minutes since takeoff ($t \ge 0$). The figure shows the graph of $s(t)$.

1. a) Determine the velocity function $v(t)$ and the acceleration function $a(t)$ of the drone.

1. b) At what instant does the drone change its direction of motion?

1. c) Calculate the exact instants at which the acceleration is zero. What physical interpretation do these points have with respect to the drone's velocity?

Consider the function $f(x) = x^4 + ax^3 + bx^2 + cx + d$. Determine the coefficients $a, b, c$ and $d$ knowing that the following conditions are simultaneously satisfied:

1. The graph of $f(x)$ passes through the origin.

2. The function has a point of inflection at $x = 1$.

3. The tangent line to the curve at the point with x-coordinate $x = 0$ is parallel to the line $y = -2x + 5$.

4. The value of the second derivative at $x = 0$ is $f''(0) = 6$.

The following graph corresponds to the derivative function, $f'(x)$, of a function $f(x)$ that is continuous on its entire domain $\mathbb{R}$. The graph of $f'(x)$ is a parabola that crosses the $X$-axis at $x=-2$ and $x=2$, and whose vertex is the point $(0,2)$.

Using the graph of $f'(x)$, answer the following questions about the original function $f(x)$, justifying your reasoning:

1. a) State the intervals of increase and decrease of $f(x)$.

1. b) Determine the $x$-coordinates of the relative extrema of $f(x)$ and classify them as maxima or minima.

1. c) At which value of $x$ does $f(x)$ have an inflection point? Justify your answer based on the relationship between $f''(x)$ and the slope of $f'(x)$.

We want to determine the real parameters $a$ and $b$ of a rational function of the form $f(x) = \frac{a x + b}{x - 2}$ (with $x \ne 2$) so that it simultaneously satisfies two conditions:

1. Its $n$-th derivative satisfies, for any natural number $n \ge 1$, the general expression:

1. The tangent line to the graph of $f$ at its $x$-intercept forms an angle of $135^\circ$ with the positive direction of the $OX$ axis.

1. a) Deduce by pattern recognition or recurrence the formula for the $n$-th order derivative of the elementary function $g(x) = \frac{1}{x - 2}$.
2. b) Decompose $f(x)$ into the form $f(x) = A + \frac{B}{x - 2}$ and use condition 1 to fix the value of $B$, obtaining a first relation between $a$ and $b$.
3. c) Analytically translate geometric condition 2 about the tangent line to find all pairs of values $(a, b)$ that satisfy the statement, checking the consistency of the result in each case.

To study the implicit curve given by the equation $x^3 + y^3 - 3xy = 0$, Cian tries to locate the points that have a horizontal tangent line and writes on the board:

"Step 1: I differentiate implicitly with respect to $x$, treating $y$ as $y(x)$:

Step 2: Since I am looking for horizontal tangents, the slope must vanish, so I set the numerator to zero: $y - x^2 = 0 \implies y = x^2$.

Step 3: I substitute $y = x^2$ into the curve: $x^3 + (x^2)^3 - 3x(x^2) = 0 \implies x^6 - 2x^3 = 0 \implies x^3(x^3 - 2) = 0$.

Step 4: I get two solutions: $x = 0$ (which gives the point $(0, 0)$) and $x = \sqrt[3]{2}$ (which gives the point $(\sqrt[3]{2}, \sqrt[3]{4})$). I conclude that at both points the curve has a horizontal tangent line $y'=0$ and, in particular, $(0,0)$ is a smooth relative extremum."

1. a) Identify the conceptual error made by Cian in Step 4 when analyzing the point $(0, 0)$. What happens when evaluating the expression for $y'$ at that point?
2. b) Justify why the condition $y' = 0$ in implicit differentiation is only a necessary condition and what additional check on the denominator is essential to perform.
3. c) Determine with justification whether a horizontal tangent line actually exists at the point $P(\sqrt[3]{2}, \sqrt[3]{4})$ and write its equation.